Q32P - Mathematical Methods in Physical Sciences

Question

Find each of the following in the x + iy form and check your answers by computer.

$\mathbf{cosh}\left(\frac{\mathrm{i\pi }}{2}- \ln 3\right)$

Step-by-Step Solution

Answer

The form of the given equation $\cosh \left(\frac{i \mathrm{\pi }}{2}-\ln 3\right) \mathrm{is} - \frac{4i}{3}$.

Step 1: Given Information.

The given expression is $\left(\frac{i \mathrm{\pi }}{2}-\ln 3\right)$.

Step 2: Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x + iy in which x is the real part and y is the imaginary part.

Step 3: Write in the form of .

Use the formula $\cosh (\mathrm{x}+\mathrm{y})=\cosh \mathrm{x} \cosh \mathrm{y}+ \sinh \mathrm{x} \sinh \mathrm{y}$to break the given question.

 

Consider the following values.

         $$\begin{gathered} x=-\ln 3 \\ y=\frac{\mathrm{\pi }}{2} \\ \cosh (\mathrm{x}+\mathrm{y})=\mathrm{coshx} \cosh \mathrm{y}+\sinh \mathrm{x} \sinh \mathrm{y} \end{gathered}$$

 

Put the values.

$\cosh \left(\frac{\mathrm{i} \mathrm{\pi }}{2}- \ln 3\right)= \cosh (-\ln 3) \cos \left(\frac{\mathrm{\pi }}{2}\right)+\mathrm{i} \sin \mathrm{h} (-\ln 3) \sinh \left(\frac{\mathrm{\pi }}{2}\right)$

 

Therefore, the x + iy form of the given equation $\cosh \left(\frac{i \mathrm{\pi }}{2}-\ln 3\right) is \frac{-4i}{3}$.